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We introduce an $A_\infty$-algebra structure on the Hochschild cohomology ofthe endomorphism bimodule of a finite-dimensional representation of anassociative algebra. We prove that this structure determines a presentation fornon-commutative deformations of the representation. From this, we deducepresentations of universal deformation rings of Galois representations. Inturn, we apply these presentations in order to deduce universal deformationrings of Galois pseudorepresentations, supplying a a tangent and obstructiontheory for pseudorepresentations. This generalizes the broadly used tangent andobstruction theory for Galois representations. We also give applications,calculating the ranks of certain Hecke algebras.

Presentations of non-commutative deformation rings via  $A_\infty$-algebras and applications to deformations of Galois  representations and pseudorepresentations